The bit and bridle are fine. Those were the first things I checked.
The trainer doesn't ride the horse, but he is ridden by one of her advanced students, and she feels this is good enough. I trust her judgment.
It's your daughter's safety that is the issue.
I used my OWN horses for my lessons and I mimicked what MY instructor did. HE was in charge of the lesson horses. They KNEW that if they misbehaved, he would dismount any of us, get on, discipline the horse, and then we would continue the lesson. This didn't happen every lesson, but we ALL, including the other lesson horses KNEW what a horse was in "trouble."
During my lessons, I always walked in the middle of the arena and I carried a whip. Should my horse not move out, I flicked the whip. I had few problems with my horses stopping, just not taking cues from my students bc of their inexperience.
My horses KNEW that I was the herd leader and to listened to ME, or else I would have a private training session after my lesson with that horse to fix the disobedience. After awhile those kind of sessions became fewer and fewer.
My lesson program MADE my horses manageable and we would take them to weekend CW events and there, out in the open, with sometimes not so great riders, they BEHAVED.
Get your trainer on this pony, or get a new trainer. Your money is going down a drain, your trainer is being lazy letting a student do the training, more like riding practice for the student, and one bad fall from a bolting pony and your daughter could fall and break her arm, or worse.
THIS is why ponies get their bad reputations, bc they rr little kids. =/